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  1. Is there a connection between Wittgenstein's argument against the theory of types, and the proof of Gödel's incompleteness theorem? (It seems that Gödel's proof relies on referring to symbols as numbers, whereas Wittgenstein's argument is that you can do no such thing in Russel's theory of types.)

  2. Was it really Wittgenstein's argument against type theory that changed logical positivism so drastically? If not, then what was it that happened before Gödel's proof which stopped analytic philosophy from thinking that it could axiomatize language? (It seems clear that Gödel's incompleteness theorem would have stopped this project in its tracks.)

  3. Should Wittgenstein be given some credit for Gödel's incompleteness theorem?

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    Cogito, Theory of Types, last I checked, was put forward, by Bertrand Russell, as a barricade against paradoxes; the problematic logic bomb was the eponymous Russell's paradox that was a death blow to Frege's attempts to ground math on set theory. However, I didn't realize there was a connection between types and the incompleteness theorems. Good to know.
    – Hudjefa
    Mar 17 at 14:48

3 Answers 3

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  1. Yes, there is a connection, as you point out. In the Tractatus, Wittgenstein writes:

    3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

    Gödel, as you know, proceeded to do precisely that.

  2. Wittgenstein's argument against type theory is one of many factors that changed Logical Positivism. Russell's "barber paradox" was another. If the history of Logical Positivism interests you, I'd recommend a delightful graphic novel called Logicomix which covers the territory nicely.

  3. Not really.

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    I thought Russell's type theory was intended to avoid paradoxes in set creation. Was W's argument against type theory inTLP or was it later? (the time line of the 'changes' is confusing to me?
    – Mitch
    Aug 14, 2011 at 14:00
  • The short answer is that the history of Logical Positivism is also the history of its unravelling. Russell's "barber paradox" pointed to problems with set theory, so he invented the theory of types to try to address that. Wittgenstein found problems in the theory of types (in the Tractatus, and afterwards); Gödel later developed his own attack along these lines, etc. Aug 17, 2011 at 6:52
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    I think I should point out a technicality that many might not get by reading a lot of popular references on the incompleteness theorems: Gödel sentences don't explicitly refer to themselves. In arithmetic or number theory, for example, there is no symbol or symbolic way of saying "this formula." However, the proofs of the incompleteness theorems explicitly construct a Gödel sentence, and show that it will always be logically equivalent (in effect) to our informal interpretation of "this statement is not provable within the theory."
    – anon
    Aug 22, 2011 at 0:10
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No.

Kurt Godel was interested in philosophy his whole life, his 2 favorite philosophers were Kant and Husserl. There is no historical evidence that his incompleteness theorems were influenced by Wittgenstein. I have read Godel's incompleteness academic papers, and Wittgenstein was not mentioned even once.

As a young man Godel studied a lot of elementary number theory, Russell's "Principia", and the theories of Peano and Frege. Later on in life he made several scathing remarks about Wittgenstein, saying that the latter "lost his mind". Wittgenstein once called incompleteness theorems liars paradox, and Godel got outraged by that remark because he thought that incompleteness theorems are just results in elementary number theory, having nothing to do with ancient greek sophistry.

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In reality, Gödel fails to construct a sentence that can be interpreted as self-referential. When attempting self-reference, it enters a loop, because the substitution leaves another substitution pending, in a return to infinity. But since the trick of arithmetization is in the middle, it makes this failure unprovable in the formal system. But that does not make it a "true" statement in any scientifically important sense. If instead of arithmetization we use the English translation from Spanish, we would have:

La auto sustitución de { The self-substitution of x it is not demonstrable } no es demostrable

We have an unresolved self-substitution, so the statement is incomplete. But if the system does not have rules for interpreting English, it remains unproven, but it is still a mere malformation due to infinite regression.

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    Mixing formal proofs with natural language is actually a recipe for disaster. If the defined operations of a formal system allow for the manipulation and there is a valid proof, then this sentence is true, no matter what natural language or other considerations from outside the system think about it. That is a point Gödel is making with this proof, actually: that any consideration rendering the sentence false must come from outside the formal system itself. Your "unresolved self-substitution is a problem" is a value-judgement not part of the formal system, which allows for it.
    – Philip Klöcking
    Mar 17 at 17:48
  • Getting emotionally involved when dealing with a formal problem is not conducive. There is no mixture of natural and formal language in my answer, it is all in natural language. But using both is absolutely necessary to introduce a topic, all the papers explain in natural language in the abstract what they are going to do, and how they will use formal language. In fact, Gödel himself dedicates the entire paragraph 1 of his original memoir to this task, and then expands on the interpretation, and explicitly cites the paradoxes that occur in natural language. Mar 17 at 19:48
  • From there he took his inspiration: "The analogy between this result and Richard's antinomy leaps to the eye; there is also a close relationship with the "liar" antinomy,14 since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not probable We are therefore confronted with a proposition which asserts its own unprovability." That something proven to be true may simply be that there is a poorly formed formula in the argument. Can you indicate which is the Diophantine equation from which unprovability is predicated? Mar 17 at 19:49

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